Problem $3(10$ Points$)$

One simple experiment used to characterize abrasive wear is to have a very hard cone $($assumed to be rigid$)$ cutting a groove of depth $\backslash ($ h $\backslash )$ and length $\backslash ($ s $\backslash )$ into a softer material $($see figure below$).$

While sliding, the normal load required to press the cone into the soft material to a depth of $\backslash ($ h $\backslash )$ is $\backslash ($ L$=\backslash$left$(\backslash$frac$\{1\}\{2\}\backslash$right$)$ H $\backslash$pi a$^\{2\}\backslash ),$ where $\backslash ($ a $\backslash )$ is the radius of the cone at the plane aligned with the top of the soft material, and $\backslash ($ H $\backslash )$ is the hardness of the soft material $($Note: the factor of $\backslash (1/2\backslash )$ in the equation for $\backslash ($ L $\backslash )$ is due to the fact that only the material under the front half of cone carries load during sliding$).$ As the cone slides, a groove with a triangular cross$-$section is formed in the material. Recall that Archard's law for wear is: $\backslash ($ W$=\backslash$frac$\{$K$\}\{$H$\}$ L $\backslash )$

a$)$ In this experiment, $\backslash ($ K $\backslash )$ in Archard's law only depends on geometry. Derive an expression for $\backslash ($ K $\backslash )$ in terms of the angle of the side of the cone, $\backslash (\backslash$theta $\backslash ),$ and which is independent of the depth of the groove, $\backslash ($ h $\backslash ).$

b$)$ It has been found experimentally that the Archard equation with the $\backslash ($ K $\backslash )$ you derived in $($a$)$ overestimates the amount of wear seen in actual abrasive conditions. Why might this be$?$